Procedure for aligning polarization modulator link for amplitude modulation applications Meredith N. Hutchinson,1,* Nicholas J. Frigo,1,2 and Vincent J. Urick1 1
Optical Sciences Division, Naval Research Laboratory, 4555 Overlook Ave SW, Washington, D.C. 20375, USA 2 Physics Department, US Naval Academy, 121 Blake Rd., Annapolis, Maryland 21402, USA * [email protected]
Abstract: A procedure is detailed for aligning the transmitted output states of a polarization modulated signal to the analyzer states of a polarizing discriminator in an analog photonic link. The steps in the procedure insure optimal amplitude modulation in the link. Experimental results are presented for biasing in two ways: either the DC bias on the modulator or a rotatable half-wave plate can be used. The corresponding theory is included. ©2014 Optical Society of America OCIS codes: (060.0060) Fiber optics and optical communications; (060.2360) Fiber optics links and subsystems; (260.5430) Polarization; (060.5625) Radio frequency photonics.
References and links 1. 2. 3. 4. 5. 6.
7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
A. L. Campillo, “Orthogonally polarized single sideband modulator,” Opt. Lett. 32(21), 3152–3154 (2007). T. E. Darcie, R. Paiam, A. Moye, J. D. Bull, H. Kato, and N. A. F. F. Jaeger, “Intensity-noise suppression in microwave-photonic links using polarization modulation,” IEEE Photon. Technol. Lett. 17(9), 1941–1943 (2005). X. Chen, W. Li, and J. Yao, “Microwave photonic link with improved dynamic range using a polarization modulator,” IEEE Photon. Technol. Lett. 25(14), 1373–1376 (2013). W. Li and J. Yao, “Dynamic range improvement of a microwave photonic link based on bi-directional use of a polarization modulator in a Sagnac loop,” Opt. Express 21(13), 15692–15697 (2013). W. Li, L. X. Wang, and N. H. Zhu, “Highly linear microwave photonic link using a polarization modulator in a Sagnac loop,” IEEE Photon. Technol. Lett. 26(1), 89–92 (2014). S. Pan, C. Wang, and J. Yao, “Generation of a stable and frequency-tunable microwave signal using a polarization modulator and a wavelength-fixed notch filter,” in Optical Fiber Communications/National Fiber Optic Engineers Conference, Technical Digest (Optical Society of America2009), paper JWA51, http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=05032402. X. Fu, H. Zhang, and M. Yao, “A New proposal of photonic analog-to-digital conversion based on polarization modulator and polarizer,” in Proceedings of the 15th Asia-Pacific Conference on Communications (Shanghai, China 2009) pp. 572–574. M. N. Hutchinson, J. M. Singley, V. J. Urick, S. R. Harmon, J. D. McKinney, and N. J. Frigo, “Mitigation of photodiode induced even-order distortion in photonic links with predistortion modulation,” (submitted) (2014). H. Zhang, S. Pan, M. Huang, and X. Chen, “Polarization-modulated analog photonic link with compensation of the dispersion-induced power fading,” Opt. Lett. 37(5), 866–868 (2012). A. L. Campillo and F. Bucholtz, “Chromatic dispersion effects in analog polarization-modulated links,” Appl. Opt. 45(12), 2742–2748 (2006). D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990). J. N. Damask, Polarization Optics in Telecommunications (Springer, 2005). N. J. Frigo, F. Bucholtz, and C. V. McLaughlin, “Polarization in phase modulated optical links: Jones- and generalized Stokes-space analysis,” J. Lightwave Technol. 31(9), 1503–1511 (2013). N. G. Walker and G. R. Walker, “Polarization control for coherent communications,” J. Lightwave Technol. 8(3), 438–458 (1990). J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40-Ghz electro-optic polarization modulator for fiber optic communication systems,” Proc. SPIE 5577, 133–143 (2004). F. Rahmatian, N. A. F. Jaeger, R. James, and E. Berolo, “An Ultrahigh-speed AlGaAs-GaAs polarization converter using slow-wave coplanar elecrodes,” IEEE Photon. Technol. Lett. 10(5), 675–677 (1998).
1. Introduction Polarization modulators have become increasingly useful for various applications of photonic links, such as orthogonally polarized single-sideband modulator [1], intensity noise suppression [2], third order distortion suppression [3–5], frequency doubling optoelectronic
#216357 - $15.00 USD Received 4 Jul 2014; revised 31 Jul 2014; accepted 1 Aug 2014; published 3 Oct 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024859 | OPTICS EXPRESS 24859
oscillator [6], and photonic analog-to-digital convertor [7], among others. We have previously presented the use of a polarization modulator (PolM) for second order distortion cancellation via modulator predistortion including the suppression of second order distortion across multiple octaves with three inputs [8]. Many publications discuss the use of a PolM for analog modulation but do not detail the experimental practicality by which that can be achieved [3, 5, 9]. This paper addresses not the PolM link per se, but the details of a procedure by which such links may be optimally aligned. Discussion of the use of a PolM in an analog photonic link typically describes the desired link operation and a desired alignment of the principal axis [9]. In practice this alignment when carried out can be difficult, as the state of polarization (SOP) can vary with environmental perturbations on the link. Further, the problem is more complicated than simply aligning the incoming SOP. That is, aligning the incoming SOP is insufficient for optimal performance: both the retardance eigenstate of the modulator (which can be considered as an SOP) and the zero modulation output SOP, when both are imaged after passing through the link, must be orthogonal to the discriminator’s pass axis on the Poincaré sphere. This can be accomplished using signals and components at the receiver, in concert with prescribed modulation states at the transmitter, as we show below. The purpose of this paper is to detail a method by which to align the SOP for maximum amplitude modulation using a PolM in an analog photonic link. In this paper, we assume a stable optical link. However, since the procedure is algorithmic, it may be possible to implement it in hardware. This would open the possibility that environmental changes in the link could be tracked at rates below some specified system processing rate. The paper is structured as follows. In section 2, the setup and the outputs are developed. In particular, we show that the DC bias at the PolM and the angle of the HWP produce equivalent biasing phase. This permits an additional approach to a non-linearity mitigation technique introduced earlier [8], which relies on bias control. In section 3 our proposed alignment procedure is detailed with a graphical representation on the Poincaré sphere. This alignment procedure enables a realization for, and more precise control of, the bias necessary for the mitigation technique [8], as well as a bias approach to polarization modulated links. In section 4, experimental results, using both the PolM DC bias input as well as the rotatable HWP orientation to set the operating point, are shown. A summary follows in section 5. 2. Setup and theory
Fig. 1. Setup for alignment of polarization modulator for amplitude modulation.
The setup for alignment is shown in Fig. 1. A CW laser is input to a PolM which is modulated with an RF source. The output of the PolM is a proxy for a link: all the components to the right of the PolM can be considered as residing at the link’s receiver. The modulated output is fed to two polarization controllers (PC) for greater alignment flexibility (PC1A and PC1), split with a 99:1 ratio coupler, with 1% fed to a polarization analyzer (PA) and the other 99% fed to another PC (PC2). The output of PC2 is then sent through a rotatable half wave plate (HWP) placed in front of a polarization beam splitter (PBS) which develops the polarization modulated signal. The two outputs are fed to either photodiodes or an optical power meter. The setup in Fig. 1 is necessary for the alignment procedure which will be detailed in Section #216357 - $15.00 USD Received 4 Jul 2014; revised 31 Jul 2014; accepted 1 Aug 2014; published 3 Oct 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024859 | OPTICS EXPRESS 24860
3; however the mathematical treatment is simpler [8] when not dealing with the added complication of rotation of SOPs through an SMF fiber. The following theory will detail a simple model system that demonstrates the feasibility of a dual output amplitude modulation format via a PolM using a rotating HWP for bias. We model the system with the positive frequency convention, with E0 eiωt as the field at the laser output. The basic components of an ideal system can be described with three Jones matrix operators [10–12], e − iδ ( t ) 2 J PolM (δ (t )) = 0
J QWP (θ ) =
0 e
+ iδ (t ) 2
(1a)
1 1 − i cos 2θ −i sin 2θ 2 −i sin 2θ 1 − i cos 2θ
cos 2ϕ J HWP (ϕ ) = −i sin 2ϕ
sin 2ϕ . − cos 2ϕ
(1b)
(1c)
In Eq. (1a), the polarization modulator creates a time-dependent retardance of δ ( t ) between the x (slow) and y components of the field, while Eqs. (1b) and (1c) describe quarter-wave and half-wave retarders respectively with slow axes oriented at θ and ϕ with respect to the x axis, respectively. A sine wave RF input to the system would correspond to an applied retardance of
δ ( t ) = φ b + δ ac = φ b + φ 0 sin ( Ω1t ) = φ b + π (Vrf Vπ ) sin ( Ω1t )
(2)
where φb corresponds to the modulator’s DC bias in Fig. 1. We have explicitly shown a single RF tone as the time-dependent retardance at the polarization modulator, δ a c , expressed in terms of the drive voltage and the modulator’s Vπ . For ideal operation, we assume that light is launched at 45 into the modulator and that the combined effect of the link, the three polarization controllers, and the coupler can be modeled as a quarter waveplate, also oriented at 45. Then the output can be written with Eq. (1) as: iωt E0 e 1 E1 E = J HWP (ϕ ) ⋅ J QWP (π / 4) ⋅ J PolM (δ (t )) 2 1 2 − iδ (t ) 2 − i cos ( 2ϕ ) sin ( 2ϕ ) 1 −i e = 2 sin ( 2ϕ ) − cos ( 2ϕ ) −i 1 0
1 iωt E0 e . + iδ t 2 e ( ) 1 0
(3)
Carrying out the matrix multiplications in Eq. (3), and multiplying each field by its complex conjugate, we find that the intensities in the two polarizations, each sent to a photodiode, can be expressed as: 1 + sin (φ 0 sin ( Ω1t ) + φ b + 4ϕ ) 1 + sin (δ a c + Φ b ) I1 1 + sin (δ (t) + 4ϕ ) I = I 0 1 − sin (δ (t) + 4ϕ ) = I 0 1 − sin (φ sin ( Ω t ) + φ + 4ϕ ) = I 0 1 − sin (δ + Φ ) ac b 0 1 b 2 (4)
where I 0 is the intensity at quadrature at either output port of the PBS, δ ac = φ0 sin Ω1t , and we have explicitly separated the AC and DC terms in the last equation. The result in Eq. (4) shows that the dual output amplitude modulation link via a PolM has the same sinusoidal transfer function as either an IM-DD or Φ M-ID link. That is, the RF output is developed with a sinusoidal transfer function and the interferometric bias point is
#216357 - $15.00 USD Received 4 Jul 2014; revised 31 Jul 2014; accepted 1 Aug 2014; published 3 Oct 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024859 | OPTICS EXPRESS 24861
determined by both the DC bias at the modulator and the angle of the rotating HWP: Φ = φ + 4ϕ , where Φ = 0 corresponds to ideal operation at quadrature bias. Thus, bias b
b
b
tuning can be accomplished by adjusting either φ at the modulator or ϕ at the HWP. This situation can be illustrated by using standard conventions [13], as in Fig. 2. Reading Eq. (3) from right to left, the launch SOP is at 45° to the modulator’s slow (x) axis. The DC and AC phase shifts ( φ and δ ( t ) , respectively, which comprise δ (t) in the modulator matrix) rotate the polarization toward and away from the right-circular polarization state. For our zero-span analysis, we use the same Poincare sphere at the receiver. The receiver’s QWP slow state is oriented at 45°, so it rotates all states by 90° about that axis, which is located at 90° to the x axis on the Poincare sphere. Similarly, the HWP, with slow axis oriented at ϕ with respect to the x axis, rotates states by 180° about an axis at 2ϕ with respect to the x axis. The trajectory for the SOP with maximum excursion is shown in Fig. 2 with heavy grey arcs, but consideration of the trajectory for a zero bias state, shows that it would evolve to a state 4ϕ radians from the −45° SOP. Recalling that an SOP’s field component passed in the x polarization is proportional to the cosine of half the angle between the SOP and the x axis [11–13], Fig. 2 shows that the modulator’s DC polarization bias φb , and the HWP’s orientation, ϕ , create identical biasing for Eqs. (3), (4), and thus can be lumped into Φ . b
b
ac
b
Fig. 2. Description of Eq. (3) in text. Light launched at + 45 to modulator’s slow axis experiences DC bias φ and ac bias δ ( t ) , shown here at maximum excursion. The QWP b
rotates states by 90° about the + 45° axis. A HWP, oriented at ϕ in the lab frame, rotates states by 180° about an axis at 2ϕ to the x axis on the Poincare sphere.
As previously reported, this link can be used to cancel second order photodiode distortion [8]. The form of Eq. (4) permits a small-signal expansion in δ a c which enables an analysis as in [8]. We use the standard Jacobi-Anger expansion for the intensities in Eq. (4) and separate the AC terms to second order in the phase modulation amplitude φ : 0
I1 − I dc = ( I 0 cos Φ b ) φ0 sin Ω1t +
1 4
( I 0 sin Φ b )φ0 cos 2 Ω1t + 2
(5)
Then the nonlinear relationship of the photodiode’s output photocurrent to the input intensity for port 1, for example, can be developed in a Taylor series:
I out = a0 I dc + a1 ( I1 − I dc ) + a2 ( I1 − I dc ) + ... 2
(6)
Substituting Eq. (5) into Eq. (6), we find that
#216357 - $15.00 USD Received 4 Jul 2014; revised 31 Jul 2014; accepted 1 Aug 2014; published 3 Oct 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024859 | OPTICS EXPRESS 24862
I out =
a 0 I 0 (1 + sin Φ b ) + ( a2 I 0 φ0 / 2) cos Φ b 2
2
2
(7)
+ ( a1 I 0 cos Φ b ) φ0 sin Ω1t + ( I 0 / 4)( a1 sin Φ b − 2a 2 I 0 cos Φ b ) φ0 cos 2 Ω1t + 2
2
,
for the DC, fundamental, and second harmonic terms to second order in the modulation amplitude. Equation (7) is similar to the earlier derivation [8] but includes both PolM and HWP biasing terms and carries all terms to second order. Since ideal operation occurs when the photocurrent mimics the modulation signal, setting the coefficient of the second order term to zero yields the bias condition for mitigating photodiode second order non-linearity [8]: sin Φ b 2
cos Φ b
=
2a2 I 0 a1
.
(8)
Equation (8) shows explicitly that biasing, either at the transmitter’s polarization modulator or at the receiver’s HWP, are equivalent, in the ability to mitigate 2nd order photodiode nonlinearity. The equivalence of the two biasing approaches described above relies on the formulation of Eq. (3): there must be correct alignment of the modulated polarization and the receiver’s analyzer. We address a procedure to accomplish this in the next section. 3. Alignment procedure
Section 2 illustrated that a dual output amplitude modulation link can be achieved, given ideal alignment. This section provides the steps necessary to realize that alignment experimentally. In Fig. 1, two PCs are present before the tap, permitting additional laboratory flexibility when aligning polarizations. The link and the procedure are best described on the Poincaré sphere. After modulation, the SOPs form an arc on the Poincaré sphere [13]. In traversing the link, the arc will be transformed by birefringent elements along the way, each element creating a rotation of SOPs on the sphere. This sequence of rotations is equivalent to a single net rotation, so that in the absence of impairments, the SOPs at the receiver form a displaced arc similar to the arc at the transmitter. This arc has arbitrary location and orientation with respect to the two polarization states of the PBS, which are represented by two points on opposite sides of the sphere. The goal is to arrange rotational transformations, with the PCs, such that (i) the arc of SOPs is on a great circle connecting the polarizer’s states (maximizing the modulation amplitude), and (ii) the SOP associated with zero modulation sits halfway between the two poles (setting the bias point at quadrature). For this situation, the detected output is identical to an interferometer biased at quadrature. Conventionally, states of linear polarization are displayed on the “equator” of the sphere, so if all the states are on the equator, the incoming arc has been transformed into linear SOPs with a modulated angle. The PBS can be used to find SOPs for the maximum and minima, but cannot be used to map the arc: intermediate intensities give only the “latitude” between the “poles” of the polarizer’s states. A PA (General Photonics PSGA_101-A) is needed as a reference to image the SOPs arriving at the PBS, and another polarization controller (PC2) can be used after the tap (and before the PBS) to align the states at the PBS with the states at the PA. It is important to note that all fiber must be secured such that the system is stabilized during alignment. Initially, the DC bias of the MZM is left at 0 V, and the HWP is set at 0°. The SOPs at both the PA and PBS are in arbitrary positions, as seen in Figs. 3(a) and 3(b) with the dotted arrows. In step 1, PC1 is moved such that the PA reads linear horizontal polarization (LHP) or Stokes parameters (1, 0, 0), which can be seen as the dotted arrow rotating to the solid arrow in Fig. 3(a). While this is happening, the input to the PBS also moves along a similar arc to an arbitrary SOP as in Fig. 3(b).
#216357 - $15.00 USD Received 4 Jul 2014; revised 31 Jul 2014; accepted 1 Aug 2014; published 3 Oct 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024859 | OPTICS EXPRESS 24863
Fig. 3. Poincare sphere representations of the SOP on the PA ((a) and (c)) and at the input to the PBS ((b) and (d)) for steps 1 and 2 of the alignment process. In step 1, PC1 brings arbitrary initial SOP to LHP on the PA. In step 2, PC2 brings PBS SOP to LHP.
In step 2, PC2 is adjusted to null output B, which passes linear vertical polarization (LVP), meaning that the SOP is in output A (LHP) at the PBS. This is shown in Fig. 3(d), where the dotted arrow rotates to LHP. Note that the SOP at the PA has not moved [Fig. 3(c)], since PC2 is outside its path. At this point there is a fixed relationship between the input to the PBS (more precisely, the HWP) and the PA that will not change as long as PC2 is kept stable. Furthermore, we know that when we observe LHP on the PA, there is LHP on the PBS. This correspondence, however, is only valid for those two linear SOPs. In step 3, the HWP is rotated by 22.5°, which translates to 45° of actual SOP rotation in the lab, and 90° when visualized on the Poincare sphere, as can be seen in Fig. 4(b). Note that the SOP does not change on the PA, as seen in Fig. 4(a), as the HWP is after the 99/1 ratio coupler. Thus, at this point, the solid arrow in Fig. 4(a) corresponds to + 45° linear polarization (L + 45°P) at the PBS. Here we are exploiting several advantages of the HWP: (i) its precision, (ii) its repeatability and reversibility, and (iii) the fact that it maps linear SOPs to linear SOPs.
Fig. 4. Poincare sphere representations of the SOP on the PA ((a) and (c)) and at the input to the PBS ((b) and (d)) for steps 3 and 4 of the alignment process. In step 3, HWP moves PBS SOP to + 45. In step 4, PC1 moves PBS SOP to LHP, so Z on PA corresponds to LHP at the PBS.
#216357 - $15.00 USD Received 4 Jul 2014; revised 31 Jul 2014; accepted 1 Aug 2014; published 3 Oct 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024859 | OPTICS EXPRESS 24864
In step 4 we adjust PC1, which moves both SOPs, to null output B. By virtue of the HWP in step 3, we know that the start and end points in this step correspond to linear SOPs at the input to the PBS. That is, although PC1 may not take the direct path shown, when the path is complete we know that the two points, LHP and L + 45°P at the PBS [Fig. 4(d)], correspond to point Z and LHP, respectively, on the PA [Fig. 4(c)]. Furthermore, by virtue of step 3, points on the red arcs in both figures lie on the great circles that comprise all linear SOPs, namely the desired modulated states. Properly biased, these arcs would correspond to an ideal excursion between quadrature and maximum transmission through a linear polarizer, namely a 45° degree change in linear SOPs. On the PA, point Z must lie on the “Greenwich meridian,” the great circle which includes right hand circular polarization (RCP) and L + 45°P and which may be marked on the PA. In step 5, PC1 is adjusted such that the state on the PA returns to LHP, which can be seen in Fig. 5(a). The rotation may not take the exact arc shown in Fig. 5(a), but at the end of the path, we know that the SOP going into the PA corresponds to the same L + 45°P as before at the PBS, and we know that Z corresponds to a linear SOP at the PBS (45 degrees away) by virtue of the HWP in step 3. This step, ideally, ‘undoes’ the transformation in step 3. At the end of the step, in Figs. 5(a) and 5(b) we know that the light going to the PA is LHP and the light going to the PBS is at L + 45°P, the ideal quadrature biasing point for the PBS discriminator. However, it remains necessary to insure that the rest of the modulated SOPs will map onto the desired arc. In step 6, the HWP is rotated back to 0°, which reliably rotates the input to the PBS back to LHP as seen in Fig. 5(d). Note that the PA has already been aligned to LHP in step 6 and does not move in step 6, as can been seen in Fig. 5(c). While this corresponds to a bias at an extremum, we include this step since it is an important experimental check on the procedure: by checking that the null remains in place during the steps that follow, one prevents “walking off” the desired bias point and desired arc during the procedure.
Fig. 5. Poincare sphere representations of the SOP on the PA ((a) and (c)) and at the input to the PBS ((b) and (d)) for steps 5 and 6 of the alignment process. In step 5, PC1 puts PA SOP onto LHP. In step 6, HWP is returned to zero degrees (reversing step 3).
#216357 - $15.00 USD Received 4 Jul 2014; revised 31 Jul 2014; accepted 1 Aug 2014; published 3 Oct 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024859 | OPTICS EXPRESS 24865
Fig. 6. Poincare sphere representations of the SOP on the PA ((a) and (c)) and at the input to the PBS ((b) and (d)) for steps 7 and 8 of the alignment process.
In step 7, a slow varying signal can be put on the DC bias, or the DC bias can be adjusted slowly to visualize where the modulation arc is on the PA, Fig. 6(a). The modulation amplitude to take the arc extremes to the “Greenwich meridian” on which Z sits, namely an amplitude of 90° on the Poincaré sphere. While it is unlikely that the extreme SOP will land on Z, by construction it is certain that a modulation amplitude of 90° on the sphere will put both extreme SOPs on Z’s meridian. The corresponding arc is shown at the input to the PBS in Fig. 6(b). In order to achieve optimal amplitude modulation, the SOPs should map to the linear states, corresponding to the equator at the PBS, and to the arc through points Z and LHP on the PA. In step 8, PC1 and PC1a are adjusted in order to properly align the arc: when it passes through LHP and point Z on the PA [Fig. 6(c)], all the corresponding points will lie on the equator (i.e. linear SOPs) at the PBS [Fig. 6(d)]. Two PCs are convenient in simultaneously satisfying the SOP constraint, shown as LHP in Fig. 6(c), and the arc constraint, shown ending at point Z in Fig. 6(c). The requirement for two PC’s is necessary as the use of one PC with 3 fractional waveplate paddles will allow the alignment of the required SOP, shown as LHP in Fig. 6(c), but not necessary also align the corresponding arc, which corresponds to a simultaneous alignment of a second arbitrary SOP [14]. Thus two PC’s are generally required to rotate the arc appropriately. In practice this procedure is accomplished by rotating either one or both of PC1 and PC1a and then bringing the SOP back to the starting point, LHP, and then applying a slow varying voltage to the DC bias to visualize the present location of the arc. This step is repeated until the arc is at the desired location. Incorporating step 6 permits an independent check of the alignment since the output at B is nulled. Once the arc is properly aligned, shown in Figs. 6(c) and 6(d) as solid arcs, one knows that the modulated SOPs are all linear, and all that remains is to set the bias point. If step 6 is performed, the current bias point is at an extremum: the SOP must be moved to either Z or its antipodal point for quadrature. This can be accomplished by either applying a DC bias on the modulator or by rotating the HWP. In the following section the difference between biasing with the modulator DC bias point and the rotating HWP will be detailed.
#216357 - $15.00 USD Received 4 Jul 2014; revised 31 Jul 2014; accepted 1 Aug 2014; published 3 Oct 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024859 | OPTICS EXPRESS 24866
4. Experimental analysis
Fig. 7. RF output power for f = 16.5GHz (red circles), 2 f = 33GHz (blue triangles) and photocurrent (green squares) along with the corresponding theory in black as a function of detector bias phase when using PolM DC bias. The comparison to actual DC bias voltage is bias phase (0.28π to 2.56π) and DC voltage (−17 V to 37 V).
Using the method outlined in the previous section the setup can be aligned to maximize amplitude modulation with balanced outputs much like a typical MZM. Analogous to the theory presented in Section 2, the DC bias of the PolM can also be used to tune the modulator transfer function. The DC bias to the PolM is included in δ ( t ) and this electrical drive can be used to adjust the detector’s bias phase while keeping the HWP fixed at 0°. The PolM from Versawave is a 40 Gb/s modulator with SMF output fiber. The PolM is fabricated from GaAs and has a large DC Vπ = 23.7 V [15]. The link was tested using a PD (u2T Photonics, Inc.) with a 1-tone input at 16.5 GHz. The PD has a bandwidth of 70 GHz. The expansion for Eq. (4) with a sinusoidal input predicts that the even and odd harmonic amplitudes depend sinusoidally on the detector’s bias phase, 90° apart from each other. Thus, the even harmonics should peak when the output is biased at DC photocurrent maxima and minima (coinciding with zero odd harmonic outputs) while the fundamental and odd harmonics should peak when the bias is at quadrature, when the DC photocurrent is half the maximum value. The output RF power for the fundamental, second harmonic and photocurrent are plotted along with the theoretical values as a function of output bias phase in Fig. 7. With zero bias phase the photocurrent corresponds to zero output. The plot compares the actual versus ideal sinusoidal behavior. The bias phase is calculated using the DC value for Vπ . The theoretical curves are calculated assuming a1 = 1 since the PD is known to be linear up to 67 GHz. Distortion of the photocurrent from the expected dependence clearly shows non-sinusoidal behavior up to a bias phase of 2π, which translates to an actual applied bias of 24 V. The response for fundamental and second order distortion up to this point (−17 V to 24 V, equivalent to a range of 0.28π to 2π in terms of bias phase) deviates from the expected behavior. We attribute the deviation from theoretical behavior as due to the modulator material properties. The GaAs modulator has the property that modal birefringence of the waveguide reduces the conversion efficiency at low-bias voltages and leads to the non-linear voltage vs. phase distortion which has been previously observed [16]. Thus a large bias voltage is necessary to implement this link in a regime that can be well modeled with the theory.
#216357 - $15.00 USD Received 4 Jul 2014; revised 31 Jul 2014; accepted 1 Aug 2014; published 3 Oct 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024859 | OPTICS EXPRESS 24867
Fig. 8. RF output power for f = 16.5GHz (red circles), 2 f = 33GHz (blue triangles) and photocurrent (green squares) along with the corresponding theory in black as a function of bias phase when using PolM DC bias which ranges from −17 V to 37V.
In contrast, if the DC bias is fixed at 0 V and the HWP is rotated the distorted behavior from the bias voltage can be avoided, as we show here. The use of the PolM link for distortion cancellation via the rotating HWP has already been detailed in [8]. Utilizing a rotatable HWP for bias tuning results in a well behaved sinusoidal optical transfer function. The above experiment was repeated, using the rotating HWP to provide the link’s bias phase, at the same RF frequencies. A full cycle occurs when tuning from −45° to 45° on the HWP when originally aligned at 0°: the angular change of a linear SOP incident on a HWP is double HWP’s angular change [12]. The result can be seen in Fig. 8. The bias phase is calculated via the angle of rotation, with 0° on the HWP corresponding to a bias extreme, i.e. minimum fundamental and maximum second harmonic. We first measured the transfer function via the angle on the HWP which has a resolution of 3°. The data was fit to a sinusoidal curve and proved very accurate, thus allowing us to gain precision by using the optical power on the second output as a means to back out the angle of rotation on the HWP. Additionally, the bias is far less prone to drift when compared to a MZM, allowing one to set the cancellation condition of Eq. (8) and have it remain stable so long as the fiber is fixed. The advantage of the bias tuning being located physically close to the detector is also important for any necessary bias feedback loops. 5. Conclusion
In summary, we presented a method to align a polarization modulator for use in an analog photonic link for the purpose of intensity modulation. The theory was detailed for using a polarization modulator in combination with a rotatable HWP and PBS for use as a dual output intensity modulation link, where the HWP provides bias tuning. The theory was experimentally validated. The use of the DC input to the modulator for bias tuning was also detailed and experimentally verified, and shown to be similar to, but not as precise, as tuning the HWP. Although the theory predicted that modulator bias and HWP bias are interchangeable, the HWP bias proved superior. The rotatable HWP provided advantages of stable and linear bias control, and matched the theory well for experiments with a 16.5 GHz input tone. The calibration procedure of Section 2 provides a repeatable and algorithmic way to optimize link intensity modulation and to set the bias point precisely when using a polarization modulator. The link architecture useful for antenna remoteing applications, increasing dynamic range and operating at 40 GHz with stable biasing as the HWP does not need to be adjusted so long as the fiber remains fixed.
#216357 - $15.00 USD Received 4 Jul 2014; revised 31 Jul 2014; accepted 1 Aug 2014; published 3 Oct 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024859 | OPTICS EXPRESS 24868
Optical Sciences Division, Naval Research Laboratory, 4555 Overlook Ave SW, Washington, D.C. 20375, USA 2 Physics Department, US Naval Academy, 121 Blake Rd., Annapolis, Maryland 21402, USA * [email protected]
Abstract: A procedure is detailed for aligning the transmitted output states of a polarization modulated signal to the analyzer states of a polarizing discriminator in an analog photonic link. The steps in the procedure insure optimal amplitude modulation in the link. Experimental results are presented for biasing in two ways: either the DC bias on the modulator or a rotatable half-wave plate can be used. The corresponding theory is included. ©2014 Optical Society of America OCIS codes: (060.0060) Fiber optics and optical communications; (060.2360) Fiber optics links and subsystems; (260.5430) Polarization; (060.5625) Radio frequency photonics.
References and links 1. 2. 3. 4. 5. 6.
7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
A. L. Campillo, “Orthogonally polarized single sideband modulator,” Opt. Lett. 32(21), 3152–3154 (2007). T. E. Darcie, R. Paiam, A. Moye, J. D. Bull, H. Kato, and N. A. F. F. Jaeger, “Intensity-noise suppression in microwave-photonic links using polarization modulation,” IEEE Photon. Technol. Lett. 17(9), 1941–1943 (2005). X. Chen, W. Li, and J. Yao, “Microwave photonic link with improved dynamic range using a polarization modulator,” IEEE Photon. Technol. Lett. 25(14), 1373–1376 (2013). W. Li and J. Yao, “Dynamic range improvement of a microwave photonic link based on bi-directional use of a polarization modulator in a Sagnac loop,” Opt. Express 21(13), 15692–15697 (2013). W. Li, L. X. Wang, and N. H. Zhu, “Highly linear microwave photonic link using a polarization modulator in a Sagnac loop,” IEEE Photon. Technol. Lett. 26(1), 89–92 (2014). S. Pan, C. Wang, and J. Yao, “Generation of a stable and frequency-tunable microwave signal using a polarization modulator and a wavelength-fixed notch filter,” in Optical Fiber Communications/National Fiber Optic Engineers Conference, Technical Digest (Optical Society of America2009), paper JWA51, http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=05032402. X. Fu, H. Zhang, and M. Yao, “A New proposal of photonic analog-to-digital conversion based on polarization modulator and polarizer,” in Proceedings of the 15th Asia-Pacific Conference on Communications (Shanghai, China 2009) pp. 572–574. M. N. Hutchinson, J. M. Singley, V. J. Urick, S. R. Harmon, J. D. McKinney, and N. J. Frigo, “Mitigation of photodiode induced even-order distortion in photonic links with predistortion modulation,” (submitted) (2014). H. Zhang, S. Pan, M. Huang, and X. Chen, “Polarization-modulated analog photonic link with compensation of the dispersion-induced power fading,” Opt. Lett. 37(5), 866–868 (2012). A. L. Campillo and F. Bucholtz, “Chromatic dispersion effects in analog polarization-modulated links,” Appl. Opt. 45(12), 2742–2748 (2006). D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990). J. N. Damask, Polarization Optics in Telecommunications (Springer, 2005). N. J. Frigo, F. Bucholtz, and C. V. McLaughlin, “Polarization in phase modulated optical links: Jones- and generalized Stokes-space analysis,” J. Lightwave Technol. 31(9), 1503–1511 (2013). N. G. Walker and G. R. Walker, “Polarization control for coherent communications,” J. Lightwave Technol. 8(3), 438–458 (1990). J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40-Ghz electro-optic polarization modulator for fiber optic communication systems,” Proc. SPIE 5577, 133–143 (2004). F. Rahmatian, N. A. F. Jaeger, R. James, and E. Berolo, “An Ultrahigh-speed AlGaAs-GaAs polarization converter using slow-wave coplanar elecrodes,” IEEE Photon. Technol. Lett. 10(5), 675–677 (1998).
1. Introduction Polarization modulators have become increasingly useful for various applications of photonic links, such as orthogonally polarized single-sideband modulator [1], intensity noise suppression [2], third order distortion suppression [3–5], frequency doubling optoelectronic
#216357 - $15.00 USD Received 4 Jul 2014; revised 31 Jul 2014; accepted 1 Aug 2014; published 3 Oct 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024859 | OPTICS EXPRESS 24859
oscillator [6], and photonic analog-to-digital convertor [7], among others. We have previously presented the use of a polarization modulator (PolM) for second order distortion cancellation via modulator predistortion including the suppression of second order distortion across multiple octaves with three inputs [8]. Many publications discuss the use of a PolM for analog modulation but do not detail the experimental practicality by which that can be achieved [3, 5, 9]. This paper addresses not the PolM link per se, but the details of a procedure by which such links may be optimally aligned. Discussion of the use of a PolM in an analog photonic link typically describes the desired link operation and a desired alignment of the principal axis [9]. In practice this alignment when carried out can be difficult, as the state of polarization (SOP) can vary with environmental perturbations on the link. Further, the problem is more complicated than simply aligning the incoming SOP. That is, aligning the incoming SOP is insufficient for optimal performance: both the retardance eigenstate of the modulator (which can be considered as an SOP) and the zero modulation output SOP, when both are imaged after passing through the link, must be orthogonal to the discriminator’s pass axis on the Poincaré sphere. This can be accomplished using signals and components at the receiver, in concert with prescribed modulation states at the transmitter, as we show below. The purpose of this paper is to detail a method by which to align the SOP for maximum amplitude modulation using a PolM in an analog photonic link. In this paper, we assume a stable optical link. However, since the procedure is algorithmic, it may be possible to implement it in hardware. This would open the possibility that environmental changes in the link could be tracked at rates below some specified system processing rate. The paper is structured as follows. In section 2, the setup and the outputs are developed. In particular, we show that the DC bias at the PolM and the angle of the HWP produce equivalent biasing phase. This permits an additional approach to a non-linearity mitigation technique introduced earlier [8], which relies on bias control. In section 3 our proposed alignment procedure is detailed with a graphical representation on the Poincaré sphere. This alignment procedure enables a realization for, and more precise control of, the bias necessary for the mitigation technique [8], as well as a bias approach to polarization modulated links. In section 4, experimental results, using both the PolM DC bias input as well as the rotatable HWP orientation to set the operating point, are shown. A summary follows in section 5. 2. Setup and theory
Fig. 1. Setup for alignment of polarization modulator for amplitude modulation.
The setup for alignment is shown in Fig. 1. A CW laser is input to a PolM which is modulated with an RF source. The output of the PolM is a proxy for a link: all the components to the right of the PolM can be considered as residing at the link’s receiver. The modulated output is fed to two polarization controllers (PC) for greater alignment flexibility (PC1A and PC1), split with a 99:1 ratio coupler, with 1% fed to a polarization analyzer (PA) and the other 99% fed to another PC (PC2). The output of PC2 is then sent through a rotatable half wave plate (HWP) placed in front of a polarization beam splitter (PBS) which develops the polarization modulated signal. The two outputs are fed to either photodiodes or an optical power meter. The setup in Fig. 1 is necessary for the alignment procedure which will be detailed in Section #216357 - $15.00 USD Received 4 Jul 2014; revised 31 Jul 2014; accepted 1 Aug 2014; published 3 Oct 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024859 | OPTICS EXPRESS 24860
3; however the mathematical treatment is simpler [8] when not dealing with the added complication of rotation of SOPs through an SMF fiber. The following theory will detail a simple model system that demonstrates the feasibility of a dual output amplitude modulation format via a PolM using a rotating HWP for bias. We model the system with the positive frequency convention, with E0 eiωt as the field at the laser output. The basic components of an ideal system can be described with three Jones matrix operators [10–12], e − iδ ( t ) 2 J PolM (δ (t )) = 0
J QWP (θ ) =
0 e
+ iδ (t ) 2
(1a)
1 1 − i cos 2θ −i sin 2θ 2 −i sin 2θ 1 − i cos 2θ
cos 2ϕ J HWP (ϕ ) = −i sin 2ϕ
sin 2ϕ . − cos 2ϕ
(1b)
(1c)
In Eq. (1a), the polarization modulator creates a time-dependent retardance of δ ( t ) between the x (slow) and y components of the field, while Eqs. (1b) and (1c) describe quarter-wave and half-wave retarders respectively with slow axes oriented at θ and ϕ with respect to the x axis, respectively. A sine wave RF input to the system would correspond to an applied retardance of
δ ( t ) = φ b + δ ac = φ b + φ 0 sin ( Ω1t ) = φ b + π (Vrf Vπ ) sin ( Ω1t )
(2)
where φb corresponds to the modulator’s DC bias in Fig. 1. We have explicitly shown a single RF tone as the time-dependent retardance at the polarization modulator, δ a c , expressed in terms of the drive voltage and the modulator’s Vπ . For ideal operation, we assume that light is launched at 45 into the modulator and that the combined effect of the link, the three polarization controllers, and the coupler can be modeled as a quarter waveplate, also oriented at 45. Then the output can be written with Eq. (1) as: iωt E0 e 1 E1 E = J HWP (ϕ ) ⋅ J QWP (π / 4) ⋅ J PolM (δ (t )) 2 1 2 − iδ (t ) 2 − i cos ( 2ϕ ) sin ( 2ϕ ) 1 −i e = 2 sin ( 2ϕ ) − cos ( 2ϕ ) −i 1 0
1 iωt E0 e . + iδ t 2 e ( ) 1 0
(3)
Carrying out the matrix multiplications in Eq. (3), and multiplying each field by its complex conjugate, we find that the intensities in the two polarizations, each sent to a photodiode, can be expressed as: 1 + sin (φ 0 sin ( Ω1t ) + φ b + 4ϕ ) 1 + sin (δ a c + Φ b ) I1 1 + sin (δ (t) + 4ϕ ) I = I 0 1 − sin (δ (t) + 4ϕ ) = I 0 1 − sin (φ sin ( Ω t ) + φ + 4ϕ ) = I 0 1 − sin (δ + Φ ) ac b 0 1 b 2 (4)
where I 0 is the intensity at quadrature at either output port of the PBS, δ ac = φ0 sin Ω1t , and we have explicitly separated the AC and DC terms in the last equation. The result in Eq. (4) shows that the dual output amplitude modulation link via a PolM has the same sinusoidal transfer function as either an IM-DD or Φ M-ID link. That is, the RF output is developed with a sinusoidal transfer function and the interferometric bias point is
#216357 - $15.00 USD Received 4 Jul 2014; revised 31 Jul 2014; accepted 1 Aug 2014; published 3 Oct 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024859 | OPTICS EXPRESS 24861
determined by both the DC bias at the modulator and the angle of the rotating HWP: Φ = φ + 4ϕ , where Φ = 0 corresponds to ideal operation at quadrature bias. Thus, bias b
b
b
tuning can be accomplished by adjusting either φ at the modulator or ϕ at the HWP. This situation can be illustrated by using standard conventions [13], as in Fig. 2. Reading Eq. (3) from right to left, the launch SOP is at 45° to the modulator’s slow (x) axis. The DC and AC phase shifts ( φ and δ ( t ) , respectively, which comprise δ (t) in the modulator matrix) rotate the polarization toward and away from the right-circular polarization state. For our zero-span analysis, we use the same Poincare sphere at the receiver. The receiver’s QWP slow state is oriented at 45°, so it rotates all states by 90° about that axis, which is located at 90° to the x axis on the Poincare sphere. Similarly, the HWP, with slow axis oriented at ϕ with respect to the x axis, rotates states by 180° about an axis at 2ϕ with respect to the x axis. The trajectory for the SOP with maximum excursion is shown in Fig. 2 with heavy grey arcs, but consideration of the trajectory for a zero bias state, shows that it would evolve to a state 4ϕ radians from the −45° SOP. Recalling that an SOP’s field component passed in the x polarization is proportional to the cosine of half the angle between the SOP and the x axis [11–13], Fig. 2 shows that the modulator’s DC polarization bias φb , and the HWP’s orientation, ϕ , create identical biasing for Eqs. (3), (4), and thus can be lumped into Φ . b
b
ac
b
Fig. 2. Description of Eq. (3) in text. Light launched at + 45 to modulator’s slow axis experiences DC bias φ and ac bias δ ( t ) , shown here at maximum excursion. The QWP b
rotates states by 90° about the + 45° axis. A HWP, oriented at ϕ in the lab frame, rotates states by 180° about an axis at 2ϕ to the x axis on the Poincare sphere.
As previously reported, this link can be used to cancel second order photodiode distortion [8]. The form of Eq. (4) permits a small-signal expansion in δ a c which enables an analysis as in [8]. We use the standard Jacobi-Anger expansion for the intensities in Eq. (4) and separate the AC terms to second order in the phase modulation amplitude φ : 0
I1 − I dc = ( I 0 cos Φ b ) φ0 sin Ω1t +
1 4
( I 0 sin Φ b )φ0 cos 2 Ω1t + 2
(5)
Then the nonlinear relationship of the photodiode’s output photocurrent to the input intensity for port 1, for example, can be developed in a Taylor series:
I out = a0 I dc + a1 ( I1 − I dc ) + a2 ( I1 − I dc ) + ... 2
(6)
Substituting Eq. (5) into Eq. (6), we find that
#216357 - $15.00 USD Received 4 Jul 2014; revised 31 Jul 2014; accepted 1 Aug 2014; published 3 Oct 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024859 | OPTICS EXPRESS 24862
I out =
a 0 I 0 (1 + sin Φ b ) + ( a2 I 0 φ0 / 2) cos Φ b 2
2
2
(7)
+ ( a1 I 0 cos Φ b ) φ0 sin Ω1t + ( I 0 / 4)( a1 sin Φ b − 2a 2 I 0 cos Φ b ) φ0 cos 2 Ω1t + 2
2
,
for the DC, fundamental, and second harmonic terms to second order in the modulation amplitude. Equation (7) is similar to the earlier derivation [8] but includes both PolM and HWP biasing terms and carries all terms to second order. Since ideal operation occurs when the photocurrent mimics the modulation signal, setting the coefficient of the second order term to zero yields the bias condition for mitigating photodiode second order non-linearity [8]: sin Φ b 2
cos Φ b
=
2a2 I 0 a1
.
(8)
Equation (8) shows explicitly that biasing, either at the transmitter’s polarization modulator or at the receiver’s HWP, are equivalent, in the ability to mitigate 2nd order photodiode nonlinearity. The equivalence of the two biasing approaches described above relies on the formulation of Eq. (3): there must be correct alignment of the modulated polarization and the receiver’s analyzer. We address a procedure to accomplish this in the next section. 3. Alignment procedure
Section 2 illustrated that a dual output amplitude modulation link can be achieved, given ideal alignment. This section provides the steps necessary to realize that alignment experimentally. In Fig. 1, two PCs are present before the tap, permitting additional laboratory flexibility when aligning polarizations. The link and the procedure are best described on the Poincaré sphere. After modulation, the SOPs form an arc on the Poincaré sphere [13]. In traversing the link, the arc will be transformed by birefringent elements along the way, each element creating a rotation of SOPs on the sphere. This sequence of rotations is equivalent to a single net rotation, so that in the absence of impairments, the SOPs at the receiver form a displaced arc similar to the arc at the transmitter. This arc has arbitrary location and orientation with respect to the two polarization states of the PBS, which are represented by two points on opposite sides of the sphere. The goal is to arrange rotational transformations, with the PCs, such that (i) the arc of SOPs is on a great circle connecting the polarizer’s states (maximizing the modulation amplitude), and (ii) the SOP associated with zero modulation sits halfway between the two poles (setting the bias point at quadrature). For this situation, the detected output is identical to an interferometer biased at quadrature. Conventionally, states of linear polarization are displayed on the “equator” of the sphere, so if all the states are on the equator, the incoming arc has been transformed into linear SOPs with a modulated angle. The PBS can be used to find SOPs for the maximum and minima, but cannot be used to map the arc: intermediate intensities give only the “latitude” between the “poles” of the polarizer’s states. A PA (General Photonics PSGA_101-A) is needed as a reference to image the SOPs arriving at the PBS, and another polarization controller (PC2) can be used after the tap (and before the PBS) to align the states at the PBS with the states at the PA. It is important to note that all fiber must be secured such that the system is stabilized during alignment. Initially, the DC bias of the MZM is left at 0 V, and the HWP is set at 0°. The SOPs at both the PA and PBS are in arbitrary positions, as seen in Figs. 3(a) and 3(b) with the dotted arrows. In step 1, PC1 is moved such that the PA reads linear horizontal polarization (LHP) or Stokes parameters (1, 0, 0), which can be seen as the dotted arrow rotating to the solid arrow in Fig. 3(a). While this is happening, the input to the PBS also moves along a similar arc to an arbitrary SOP as in Fig. 3(b).
#216357 - $15.00 USD Received 4 Jul 2014; revised 31 Jul 2014; accepted 1 Aug 2014; published 3 Oct 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024859 | OPTICS EXPRESS 24863
Fig. 3. Poincare sphere representations of the SOP on the PA ((a) and (c)) and at the input to the PBS ((b) and (d)) for steps 1 and 2 of the alignment process. In step 1, PC1 brings arbitrary initial SOP to LHP on the PA. In step 2, PC2 brings PBS SOP to LHP.
In step 2, PC2 is adjusted to null output B, which passes linear vertical polarization (LVP), meaning that the SOP is in output A (LHP) at the PBS. This is shown in Fig. 3(d), where the dotted arrow rotates to LHP. Note that the SOP at the PA has not moved [Fig. 3(c)], since PC2 is outside its path. At this point there is a fixed relationship between the input to the PBS (more precisely, the HWP) and the PA that will not change as long as PC2 is kept stable. Furthermore, we know that when we observe LHP on the PA, there is LHP on the PBS. This correspondence, however, is only valid for those two linear SOPs. In step 3, the HWP is rotated by 22.5°, which translates to 45° of actual SOP rotation in the lab, and 90° when visualized on the Poincare sphere, as can be seen in Fig. 4(b). Note that the SOP does not change on the PA, as seen in Fig. 4(a), as the HWP is after the 99/1 ratio coupler. Thus, at this point, the solid arrow in Fig. 4(a) corresponds to + 45° linear polarization (L + 45°P) at the PBS. Here we are exploiting several advantages of the HWP: (i) its precision, (ii) its repeatability and reversibility, and (iii) the fact that it maps linear SOPs to linear SOPs.
Fig. 4. Poincare sphere representations of the SOP on the PA ((a) and (c)) and at the input to the PBS ((b) and (d)) for steps 3 and 4 of the alignment process. In step 3, HWP moves PBS SOP to + 45. In step 4, PC1 moves PBS SOP to LHP, so Z on PA corresponds to LHP at the PBS.
#216357 - $15.00 USD Received 4 Jul 2014; revised 31 Jul 2014; accepted 1 Aug 2014; published 3 Oct 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024859 | OPTICS EXPRESS 24864
In step 4 we adjust PC1, which moves both SOPs, to null output B. By virtue of the HWP in step 3, we know that the start and end points in this step correspond to linear SOPs at the input to the PBS. That is, although PC1 may not take the direct path shown, when the path is complete we know that the two points, LHP and L + 45°P at the PBS [Fig. 4(d)], correspond to point Z and LHP, respectively, on the PA [Fig. 4(c)]. Furthermore, by virtue of step 3, points on the red arcs in both figures lie on the great circles that comprise all linear SOPs, namely the desired modulated states. Properly biased, these arcs would correspond to an ideal excursion between quadrature and maximum transmission through a linear polarizer, namely a 45° degree change in linear SOPs. On the PA, point Z must lie on the “Greenwich meridian,” the great circle which includes right hand circular polarization (RCP) and L + 45°P and which may be marked on the PA. In step 5, PC1 is adjusted such that the state on the PA returns to LHP, which can be seen in Fig. 5(a). The rotation may not take the exact arc shown in Fig. 5(a), but at the end of the path, we know that the SOP going into the PA corresponds to the same L + 45°P as before at the PBS, and we know that Z corresponds to a linear SOP at the PBS (45 degrees away) by virtue of the HWP in step 3. This step, ideally, ‘undoes’ the transformation in step 3. At the end of the step, in Figs. 5(a) and 5(b) we know that the light going to the PA is LHP and the light going to the PBS is at L + 45°P, the ideal quadrature biasing point for the PBS discriminator. However, it remains necessary to insure that the rest of the modulated SOPs will map onto the desired arc. In step 6, the HWP is rotated back to 0°, which reliably rotates the input to the PBS back to LHP as seen in Fig. 5(d). Note that the PA has already been aligned to LHP in step 6 and does not move in step 6, as can been seen in Fig. 5(c). While this corresponds to a bias at an extremum, we include this step since it is an important experimental check on the procedure: by checking that the null remains in place during the steps that follow, one prevents “walking off” the desired bias point and desired arc during the procedure.
Fig. 5. Poincare sphere representations of the SOP on the PA ((a) and (c)) and at the input to the PBS ((b) and (d)) for steps 5 and 6 of the alignment process. In step 5, PC1 puts PA SOP onto LHP. In step 6, HWP is returned to zero degrees (reversing step 3).
#216357 - $15.00 USD Received 4 Jul 2014; revised 31 Jul 2014; accepted 1 Aug 2014; published 3 Oct 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024859 | OPTICS EXPRESS 24865
Fig. 6. Poincare sphere representations of the SOP on the PA ((a) and (c)) and at the input to the PBS ((b) and (d)) for steps 7 and 8 of the alignment process.
In step 7, a slow varying signal can be put on the DC bias, or the DC bias can be adjusted slowly to visualize where the modulation arc is on the PA, Fig. 6(a). The modulation amplitude to take the arc extremes to the “Greenwich meridian” on which Z sits, namely an amplitude of 90° on the Poincaré sphere. While it is unlikely that the extreme SOP will land on Z, by construction it is certain that a modulation amplitude of 90° on the sphere will put both extreme SOPs on Z’s meridian. The corresponding arc is shown at the input to the PBS in Fig. 6(b). In order to achieve optimal amplitude modulation, the SOPs should map to the linear states, corresponding to the equator at the PBS, and to the arc through points Z and LHP on the PA. In step 8, PC1 and PC1a are adjusted in order to properly align the arc: when it passes through LHP and point Z on the PA [Fig. 6(c)], all the corresponding points will lie on the equator (i.e. linear SOPs) at the PBS [Fig. 6(d)]. Two PCs are convenient in simultaneously satisfying the SOP constraint, shown as LHP in Fig. 6(c), and the arc constraint, shown ending at point Z in Fig. 6(c). The requirement for two PC’s is necessary as the use of one PC with 3 fractional waveplate paddles will allow the alignment of the required SOP, shown as LHP in Fig. 6(c), but not necessary also align the corresponding arc, which corresponds to a simultaneous alignment of a second arbitrary SOP [14]. Thus two PC’s are generally required to rotate the arc appropriately. In practice this procedure is accomplished by rotating either one or both of PC1 and PC1a and then bringing the SOP back to the starting point, LHP, and then applying a slow varying voltage to the DC bias to visualize the present location of the arc. This step is repeated until the arc is at the desired location. Incorporating step 6 permits an independent check of the alignment since the output at B is nulled. Once the arc is properly aligned, shown in Figs. 6(c) and 6(d) as solid arcs, one knows that the modulated SOPs are all linear, and all that remains is to set the bias point. If step 6 is performed, the current bias point is at an extremum: the SOP must be moved to either Z or its antipodal point for quadrature. This can be accomplished by either applying a DC bias on the modulator or by rotating the HWP. In the following section the difference between biasing with the modulator DC bias point and the rotating HWP will be detailed.
#216357 - $15.00 USD Received 4 Jul 2014; revised 31 Jul 2014; accepted 1 Aug 2014; published 3 Oct 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024859 | OPTICS EXPRESS 24866
4. Experimental analysis
Fig. 7. RF output power for f = 16.5GHz (red circles), 2 f = 33GHz (blue triangles) and photocurrent (green squares) along with the corresponding theory in black as a function of detector bias phase when using PolM DC bias. The comparison to actual DC bias voltage is bias phase (0.28π to 2.56π) and DC voltage (−17 V to 37 V).
Using the method outlined in the previous section the setup can be aligned to maximize amplitude modulation with balanced outputs much like a typical MZM. Analogous to the theory presented in Section 2, the DC bias of the PolM can also be used to tune the modulator transfer function. The DC bias to the PolM is included in δ ( t ) and this electrical drive can be used to adjust the detector’s bias phase while keeping the HWP fixed at 0°. The PolM from Versawave is a 40 Gb/s modulator with SMF output fiber. The PolM is fabricated from GaAs and has a large DC Vπ = 23.7 V [15]. The link was tested using a PD (u2T Photonics, Inc.) with a 1-tone input at 16.5 GHz. The PD has a bandwidth of 70 GHz. The expansion for Eq. (4) with a sinusoidal input predicts that the even and odd harmonic amplitudes depend sinusoidally on the detector’s bias phase, 90° apart from each other. Thus, the even harmonics should peak when the output is biased at DC photocurrent maxima and minima (coinciding with zero odd harmonic outputs) while the fundamental and odd harmonics should peak when the bias is at quadrature, when the DC photocurrent is half the maximum value. The output RF power for the fundamental, second harmonic and photocurrent are plotted along with the theoretical values as a function of output bias phase in Fig. 7. With zero bias phase the photocurrent corresponds to zero output. The plot compares the actual versus ideal sinusoidal behavior. The bias phase is calculated using the DC value for Vπ . The theoretical curves are calculated assuming a1 = 1 since the PD is known to be linear up to 67 GHz. Distortion of the photocurrent from the expected dependence clearly shows non-sinusoidal behavior up to a bias phase of 2π, which translates to an actual applied bias of 24 V. The response for fundamental and second order distortion up to this point (−17 V to 24 V, equivalent to a range of 0.28π to 2π in terms of bias phase) deviates from the expected behavior. We attribute the deviation from theoretical behavior as due to the modulator material properties. The GaAs modulator has the property that modal birefringence of the waveguide reduces the conversion efficiency at low-bias voltages and leads to the non-linear voltage vs. phase distortion which has been previously observed [16]. Thus a large bias voltage is necessary to implement this link in a regime that can be well modeled with the theory.
#216357 - $15.00 USD Received 4 Jul 2014; revised 31 Jul 2014; accepted 1 Aug 2014; published 3 Oct 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024859 | OPTICS EXPRESS 24867
Fig. 8. RF output power for f = 16.5GHz (red circles), 2 f = 33GHz (blue triangles) and photocurrent (green squares) along with the corresponding theory in black as a function of bias phase when using PolM DC bias which ranges from −17 V to 37V.
In contrast, if the DC bias is fixed at 0 V and the HWP is rotated the distorted behavior from the bias voltage can be avoided, as we show here. The use of the PolM link for distortion cancellation via the rotating HWP has already been detailed in [8]. Utilizing a rotatable HWP for bias tuning results in a well behaved sinusoidal optical transfer function. The above experiment was repeated, using the rotating HWP to provide the link’s bias phase, at the same RF frequencies. A full cycle occurs when tuning from −45° to 45° on the HWP when originally aligned at 0°: the angular change of a linear SOP incident on a HWP is double HWP’s angular change [12]. The result can be seen in Fig. 8. The bias phase is calculated via the angle of rotation, with 0° on the HWP corresponding to a bias extreme, i.e. minimum fundamental and maximum second harmonic. We first measured the transfer function via the angle on the HWP which has a resolution of 3°. The data was fit to a sinusoidal curve and proved very accurate, thus allowing us to gain precision by using the optical power on the second output as a means to back out the angle of rotation on the HWP. Additionally, the bias is far less prone to drift when compared to a MZM, allowing one to set the cancellation condition of Eq. (8) and have it remain stable so long as the fiber is fixed. The advantage of the bias tuning being located physically close to the detector is also important for any necessary bias feedback loops. 5. Conclusion
In summary, we presented a method to align a polarization modulator for use in an analog photonic link for the purpose of intensity modulation. The theory was detailed for using a polarization modulator in combination with a rotatable HWP and PBS for use as a dual output intensity modulation link, where the HWP provides bias tuning. The theory was experimentally validated. The use of the DC input to the modulator for bias tuning was also detailed and experimentally verified, and shown to be similar to, but not as precise, as tuning the HWP. Although the theory predicted that modulator bias and HWP bias are interchangeable, the HWP bias proved superior. The rotatable HWP provided advantages of stable and linear bias control, and matched the theory well for experiments with a 16.5 GHz input tone. The calibration procedure of Section 2 provides a repeatable and algorithmic way to optimize link intensity modulation and to set the bias point precisely when using a polarization modulator. The link architecture useful for antenna remoteing applications, increasing dynamic range and operating at 40 GHz with stable biasing as the HWP does not need to be adjusted so long as the fiber remains fixed.
#216357 - $15.00 USD Received 4 Jul 2014; revised 31 Jul 2014; accepted 1 Aug 2014; published 3 Oct 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024859 | OPTICS EXPRESS 24868
